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Section 1.1 Integers

In mathematics symbols are used to obtain a clearer and shorter presentation. The first of these symbols is the ellipses (\(\ldots\)). When we use this symbol in mathematics, it means “continuing in this manner.” When a pattern is evident, we can use the ellipses (\(\ldots\)) to indicate that the pattern continues. We use this to define the integers.

The integers are the numbers

\begin{equation*} \ldots,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,\ldots \end{equation*}

The natural numbers, or positive integers are:

\begin{equation*} 1,2,3,4,5,6,7,\ldots \end{equation*}

The negative integers are:

\begin{equation*} \ldots,-7,-6,-5,-4,-3,-2,-1 \end{equation*}

The integer \(0\) is not considered to be positive or negative.

In the video in Figure 1.1.1 we give an introduction to the integers and statements.

Figure 1.1.1. The Integers (Part I: Definition and Statements) by Matt Farmer and Stephen Steward

Figure 1.1.2 shows the integers, natural numbers, and negative integers on the number line.

(a) The integers on the number line extend to the left and right.
(b) The natural numbers (or positive integers) on the number line extend only to the right starting at \(1\text{.}\)
(c) The negative integers on the number line extend only to the left starting at \(-1\text{.}\)
Figure 1.1.2. Number lines

Subsection 1.1.1 Comparing Integers

The symbols \(=\text{,}\) \(\ne\text{,}\) \(\lt\text{,}\) \(\le\text{,}\) \(>\text{,}\) and \(\ge\) are used to compare integers.

Table 1.1.3. Comparison symbols for integers
symbol read as
\(=\) “is equal to”
\(\ne\) “is not equal to”
\(>\) “is greater than”
\(\ge\) “is greater than or equal to”
\(\lt\) “is less than”
\(\le\) “is less than or equal to”

The first symbol in Table 1.1.3 is the equality symbol, \(=\text{.}\) Two integers are equal if they are the same integer. To indicate that two integers are not equal we use the symbol, \(\ne\text{.}\)

The other symbols compare the positions of two integers on the number line. An integer is greater than another integer if the first integer is to the right of the second integer on the number line. An integer is less than another integer if the first integer is to the left of the second integer on the number line.

We give examples of comparisons and how to read them.

  1. \(2=2\) is read “2 is equal to 2.”

  2. \(2\ne 3\) is read “\(2\) is not equal to 3.”

  3. \(3> 2\) is read “3 is greater than 2.”

  4. \(3\ge 2\) is read “3 is greater than or equal to 2.”

  5. \(2\lt 3\) is read “2 is less than 3.”

  6. \(2\le 3\) is read “2 is less than or equal to 3.”

In the Checkpoint 1.1.5 select the correct comparison operator.

Choose the symbol that yields a true statement:

-200

  • <

  • =

  • >

94

Answer.

\({\verb!<!}\)

Subsection 1.1.2 Operations

Addition, negation, subtraction, and multiplication are the basic operations of integers. We write “\(+\)” for plus, “\(-\)” for minus, and “\(\cdot\)” for times.

We give some examples of statements that involve integer operations. As we do not say “is false” we mean that all of these equality statements are true.

  1. \(2+3=5\) is read “2 plus 3 is equal to 5”

  2. \(2+0=2\) is read “2 plus 0 is equal to 2”

  3. \(2+(-2)=0\) is read “2 plus negative 2 is equal to 0”

  4. \(2-2=0\) is read “2 minus 2 is equal to 0”

  5. \(2\cdot 5=10\) is read “2 times 5 is equal to 10”

  6. \(2\cdot(-5)=-10\) is read “2 times negative 5 is equal to negative 10”

  7. \((-2)\cdot(-5)=10\) is read “negative 2 times negative 5 is equal to 10”

Multiplication of a natural number with an integer can be viewed as repeated addition.

We give examples of multiplication viewed as repeated addition.

  1. \(\displaystyle 3 \cdot 5 = 5+5+5=15\)

  2. \(\displaystyle 3 \cdot (-5) = (-5)+(-5)+(-5)=-15\)

  3. Again, we can use ellipses (\(\ldots\)) to represent a continuing pattern:

    \begin{equation*} 100 \cdot 5 =\underbrace{5+5+\ldots+5}_{100 \text{ copies of }5}=500\text{.} \end{equation*}

Defining the multiplication of two negative integers is more difficult, and we appeal to your previously acquired knowledge about integers for that. Recall that the product of two negative integers is positive.

We give examples of multiplication of integers and negative integers:

  1. \(\displaystyle 3\cdot(-5) = -15\)

  2. \(\displaystyle (-3)\cdot 5=-15\)

  3. \(\displaystyle (-3)\cdot (-5)=15\)

Subsection 1.1.3 Order of Operations

We use parentheses to indicate the order in which expressions should be executed. We evaluate the expressions in the innermost parentheses first and then work our way outwards.

We give examples for order of operations. The numbers and the operations are the same; only the grouping of the expressions given by the parentheses differs.

  1. \(\displaystyle (2+3)\cdot 4=5\cdot 4=20\)

  2. \(\displaystyle 2+(3\cdot 4)=2+12=14\)

  3. \(\displaystyle 5\cdot \left(2+(3\cdot 4)\right)=5\cdot(2+12)=5\cdot 14=70\)

  4. \(\displaystyle (5\cdot 2)+(3\cdot 4)=10+12=22\)

XKCD Presents: New Mnemonics Order of Operations: Parentheses, Exponents, Division and Multiplication, Addition and Subtraction Traditional Mnemonic: Please Excuse My Dear Aunt Sally Person having a shark delivered to his laptop. New Mnemonics: Please Email My Dad A Shark or People Expect More Drugs And Sex
Figure 1.1.10. From Mnemonics by Randall Munroe (https://xkcd.com/992).

We illustrate that the order of operations does not matter for repeated addition by computing the same sums in the order indicated by the parentheses.

  1. \(\displaystyle ((1+2)+3)+4=(3+3)+4=6+4=10\)

  2. \(\displaystyle 1+((2+3)+4)=1+(5+4)=1+9=10\)

  3. \(\displaystyle (1+2)+(3+4)=3+7=10\)

Usually we write \(1+2+3+4=10\text{.}\)

In most cases we will use parentheses to indicate the order of operations. There are other conventions for implicit order of operations (see Figure 1.1.10). One of these conventions is that multiplication is performed before addition and subtraction. We will use this convention when we feel that the additional parentheses will make it hard to read the expressions under consideration.

In the video in Figure 1.1.12 we recap the operations for the integers and give a motivation for the following section.

Figure 1.1.12. The Integers (Part 2: Operations) by Matt Farmer and Stephen Steward